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= Title2 =
= Title2 =
== Octave compression ==
== Octave stretch or compression ==
What follows is a comparison of compressed-octave 22edo tunings.
What follows is a comparison of stretched-octave 31edo tunings.


; 22edo
; EDONAME
* Step size: 54.545{{c}}, octave size: 1200.0{{c}}  
* Step size: 38.710{{c}}, octave size: 1200.0{{c}}  
Pure-octaves 22edo approximates all harmonics up to 16 within 22.3{{c}}. The optimal 13-limit [[WE]] tuning has octaves only 0.01{{c}} different from pure-octaves 22edo, and the 13-limit [[TE]] tuning has octaves only 0.08{{c}} different, so by those metrics pure-octaves 22edo might be considered already optimal. It is a good 13-limit tuning for its size.
Pure-octaves 31edo approximates all harmonics up to 16 within NNN{{c}}.
{{Harmonics in equal|22|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 22edo}}
{{Harmonics in equal|31|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONAME}}
{{Harmonics in equal|22|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 22edo (continued)}}
{{Harmonics in equal|31|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONAME (continued)}}


; [[WE|22et, 11-limit WE tuning]]  
; [[WE|31et, 13-limit WE tuning]]  
* Step size: 54.494{{c}}, octave size: 1198.9{{c}}
* Step size: 38.725{{c}}, octave size: NNN{{c}}
Compressing the octave of 22edo by around 1{{c}} results in slightly improved primes 3 and 7, but slightly worse primes 5 and 11, and a much worse 13. This approximates all harmonics up to 16 within 26.5{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this. It is a good 11-limit tuning for its size.
Stretching the octave of 31edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
{{Harmonics in cet|54.494|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 22et, 11-limit WE tuning}}
{{Harmonics in cet|38.725|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 13-limit WE tuning}}
{{Harmonics in cet|54.494|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 22et, 11-limit WE tuning (continued)}}
{{Harmonics in cet|38.725|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, SUBGROUP WE tuning (continued)}}


; [[zpi|80zpi]]  
; [[zpi|127zpi]]  
* Step size: 54.483{{c}}, octave size: 1198.6{{c}}
* Step size: 38.737{{c}}, octave size: NNN{{c}}
Compressing the octave of 22edo by around 1{{c}} results in slightly improved primes 3 and 7, but slightly worse primes 5 and 11, and a much worse 13. This approximates all harmonics up to 16 within 27.1{{c}}. The tuning 80zpi does this. It is a good 11-limit tuning for its size.
Stretching the octave of 31edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 127zpi does this.
{{Harmonics in cet|54.483|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 80zpi}}
{{Harmonics in cet|100|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 127zpi}}
{{Harmonics in cet|54.483|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 80zpi (continued)}}
{{Harmonics in cet|100|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 127zpi (continued)}}


; [[57ed6]]  
; [[WE|31et, 11-limit WE tuning]]  
* Step size: NNN{{c}}, octave size: 1197.2{{c}}
* Step size: 38.748{{c}}, octave size: NNN{{c}}
Compressing the octave of 22edo by around 3{{c}} results in greatly improved primes 3 and 7, but far worse primes 5 and 11 and a [[JND|just noticeably worse]] 2. The mapping of 13 differs from 22edo but has about the same amount of error. This approximates all harmonics up to 16 within 21.9{{c}}. With its worse 5 and 11, it only really makes sense as a [[2.3.7]] tuning (eg for [[archy]] temperament). The tuning 57ed6 does this.
_Stretching the octave of 31edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this.
{{Harmonics in equal|57|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 57ed6}}
{{Harmonics in cet|38.748|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 11-limit WE tuning}}
{{Harmonics in equal|57|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 57ed6 (continued)}}
{{Harmonics in cet|38.748|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 11-limit WE tuning (continued)}}


; [[35edt]]  
; [[111ed12]]  
* Step size: NNN{{c}}, octave size: 1195.5{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}
Compressing the octave of 22edo by around 4.5{{c}} results in greatly improved primes 3, 7 and 13, but far worse primes 5 and 11 and a [[JND|just noticeably worse]] 2. This approximates all harmonics up to 16 within 21.4{{c}}. The tunings 35edt and [[62ed7]] both do this. This extends 57ed6's 2.3.7 tuning into a 2.3.7.13 [[subgroup]] tuning.
Stretching the octave of 31edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 111ed12 does this.
{{Harmonics in equal|35|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 35edt}}
{{Harmonics in equal|111|12|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 111ed12}}
{{Harmonics in equal|35|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 35edt (continued)}}
{{Harmonics in equal|111|12|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 111ed12 (continued)}}
 
; [[80ed6]]  
* Step size: NNN{{c}}, octave size: NNN{{c}}
Stretching the octave of 31edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 80ed6 does this.
{{Harmonics in equal|80|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6}}
{{Harmonics in equal|80|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6 (continued)}}
 
; [[25ed7/4]]  
* Step size: NNN{{c}}, octave size: NNN{{c}}
Stretching the octave of 31edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 25ed7/4 does this.
{{Harmonics in equal|25|7|4|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 25ed7/4}}
{{Harmonics in equal|25|7|4|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 25ed7/4 (continued)}}

Revision as of 05:06, 24 August 2025

Title1

Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -4.1 -8.5 -8.2 +4.1 -12.6 +19.5 -12.3 -16.9 +0.0 +34.3 -16.7
Relative (%) -4.1 -8.5 -8.2 +4.1 -12.6 +19.6 -12.4 -17.0 +0.0 +34.4 -16.7
Steps
(reduced) 		12
(12) 		19
(19) 		24
(24) 		28
(28) 		31
(31) 		34
(34) 		36
(36) 		38
(38) 		40
(0) 		42
(2) 		43
(3)
Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +3.4 +3.4 +6.7 +21.5 +6.7 +40.7 +10.1 +6.7 +24.9 -39.9 +10.1
Relative (%) +3.3 +3.3 +6.7 +21.4 +6.7 +40.6 +10.0 +6.7 +24.8 -39.8 +10.0
Steps
(reduced) 		12
(5) 		19
(5) 		24
(3) 		28
(0) 		31
(3) 		34
(6) 		36
(1) 		38
(3) 		40
(5) 		41
(6) 		43
(1)
Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.2 +0.0 +2.5 +16.6 +1.2 +34.7 +3.7 +0.0 +17.8 -47.1 +2.5
Relative (%) +1.2 +0.0 +2.5 +16.6 +1.2 +34.6 +3.7 +0.0 +17.8 -47.1 +2.5
Steps
(reduced) 		12
(12) 		19
(0) 		24
(5) 		28
(9) 		31
(12) 		34
(15) 		36
(17) 		38
(0) 		40
(2) 		41
(3) 		43
(5)
Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.8 -0.8 +1.5 +15.5 +0.0 +33.3 +2.3 -1.5 +16.2 -48.7 +0.8
Relative (%) +0.8 -0.8 +1.5 +15.4 +0.0 +33.3 +2.3 -1.5 +16.2 -48.7 +0.8
Steps
(reduced) 		12
(12) 		19
(19) 		24
(24) 		28
(28) 		31
(0) 		34
(3) 		36
(5) 		38
(7) 		40
(9) 		41
(10) 		43
(12)

Title2

Octave stretch or compression

What follows is a comparison of stretched-octave 31edo tunings.

EDONAME
  • Step size: 38.710 ¢, octave size: 1200.0 ¢

Pure-octaves 31edo approximates all harmonics up to 16 within NNN ¢.

Approximation of harmonics in EDONAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -5.2 +0.0 +0.8 -5.2 -1.1 +0.0 -10.4 +0.8 -9.4 -5.2
Relative (%) +0.0 -13.4 +0.0 +2.0 -13.4 -2.8 +0.0 -26.8 +2.0 -24.2 -13.4
Steps
(reduced) 		31
(0) 		49
(18) 		62
(0) 		72
(10) 		80
(18) 		87
(25) 		93
(0) 		98
(5) 		103
(10) 		107
(14) 		111
(18)
Approximation of harmonics in EDONAME (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +11.1 -1.1 -4.4 +0.0 +11.2 -10.4 +12.2 +0.8 -6.3 -9.4 -8.9 -5.2
Relative (%) +28.6 -2.8 -11.4 +0.0 +28.9 -26.8 +31.4 +2.0 -16.2 -24.2 -23.0 -13.4
Steps
(reduced) 		115
(22) 		118
(25) 		121
(28) 		124
(0) 		127
(3) 		129
(5) 		132
(8) 		134
(10) 		136
(12) 		138
(14) 		140
(16) 		142
(18)
31et, 13-limit WE tuning
  • Step size: 38.725 ¢, octave size: NNN ¢

Stretching the octave of 31edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its 13-limit WE tuning and 13-limit TE tuning both do this.

Approximation of harmonics in 31et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.5 -4.4 +0.9 +1.9 -4.0 +0.2 +1.4 -8.9 +2.4 -7.7 -3.5
Relative (%) +1.2 -11.4 +2.5 +4.9 -10.2 +0.6 +3.7 -22.9 +6.1 -20.0 -9.0
Step 31 49 62 72 80 87 93 98 103 107 111
Approximation of harmonics in 31et, SUBGROUP WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +12.8 +0.7 -2.5 +1.9 +13.1 -8.4 +14.2 +2.8 -4.2 -7.3 -6.8 -3.0
Relative (%) +33.2 +1.9 -6.6 +4.9 +33.9 -21.7 +36.6 +7.3 -10.8 -18.8 -17.5 -7.8
Step 115 118 121 124 127 129 132 134 136 138 140 142
127zpi
  • Step size: 38.737 ¢, octave size: NNN ¢

Stretching the octave of 31edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 127zpi does this.

Approximation of harmonics in 127zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Relative (%) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Step 12 19 24 28 31 34 36 38 40 42 43
Approximation of harmonics in 127zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Relative (%) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Step 44 46 47 48 49 50 51 52 53 54 54 55
31et, 11-limit WE tuning
  • Step size: 38.748 ¢, octave size: NNN ¢

_Stretching the octave of 31edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its 11-limit WE tuning and 11-limit TE tuning both do this.

Approximation of harmonics in 31et, 11-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.2 -3.3 +2.4 +3.5 -2.1 +2.3 +3.6 -6.6 +4.7 -5.3 -0.9
Relative (%) +3.1 -8.5 +6.1 +9.1 -5.5 +5.8 +9.2 -17.0 +12.2 -13.6 -2.4
Step 31 49 62 72 80 87 93 98 103 107 111
Approximation of harmonics in 31et, 11-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +15.5 +3.4 +0.2 +4.8 +16.0 -5.4 +17.2 +5.9 -1.1 -4.1 -3.6 +0.3
Relative (%) +40.0 +8.9 +0.6 +12.3 +41.4 -14.0 +44.4 +15.3 -2.7 -10.6 -9.2 +0.7
Step 115 118 121 124 127 129 132 134 136 138 140 142
111ed12
  • Step size: NNN ¢, octave size: NNN ¢

Stretching the octave of 31edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 111ed12 does this.

Approximation of harmonics in 111ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.4 -2.9 +2.9 +4.1 -1.4 +3.0 +4.3 -5.8 +5.6 -4.4 +0.0
Relative (%) +3.7 -7.5 +7.5 +10.7 -3.7 +7.7 +11.2 -14.9 +14.4 -11.3 +0.0
Steps
(reduced) 		31
(31) 		49
(49) 		62
(62) 		72
(72) 		80
(80) 		87
(87) 		93
(93) 		98
(98) 		103
(103) 		107
(107) 		111
(0)
Approximation of harmonics in 111ed12 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +16.5 +4.4 +1.2 +5.8 +17.1 -4.3 +18.3 +7.0 +0.1 -2.9 -2.4 +1.4
Relative (%) +42.5 +11.4 +3.2 +14.9 +44.1 -11.2 +47.3 +18.2 +0.2 -7.6 -6.2 +3.7
Steps
(reduced) 		115
(4) 		118
(7) 		121
(10) 		124
(13) 		127
(16) 		129
(18) 		132
(21) 		134
(23) 		136
(25) 		138
(27) 		140
(29) 		142
(31)
80ed6
  • Step size: NNN ¢, octave size: NNN ¢

Stretching the octave of 31edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 80ed6 does this.

Approximation of harmonics in 80ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +2.0 -2.0 +4.0 +5.4 +0.0 +4.6 +6.0 -4.0 +7.5 -2.5 +2.0
Relative (%) +5.2 -5.2 +10.4 +14.0 +0.0 +11.7 +15.5 -10.4 +19.2 -6.3 +5.2
Steps
(reduced) 		31
(31) 		49
(49) 		62
(62) 		72
(72) 		80
(0) 		87
(7) 		93
(13) 		98
(18) 		103
(23) 		107
(27) 		111
(31)
Approximation of harmonics in 80ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +18.5 +6.6 +3.4 +8.0 -19.4 -2.0 -18.1 +9.5 +2.5 -0.4 +0.1 +4.0
Relative (%) +47.8 +16.9 +8.9 +20.7 -50.0 -5.2 -46.6 +24.4 +6.6 -1.1 +0.4 +10.4
Steps
(reduced) 		115
(35) 		118
(38) 		121
(41) 		124
(44) 		126
(46) 		129
(49) 		131
(51) 		134
(54) 		136
(56) 		138
(58) 		140
(60) 		142
(62)
25ed7/4
  • Step size: NNN ¢, octave size: NNN ¢

Stretching the octave of 31edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 25ed7/4 does this.

Approximation of harmonics in 25ed7/4
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.3 -3.1 +2.7 +3.9 -1.7 +2.7 +4.0 -6.1 +5.2 -4.7 -0.4
Relative (%) +3.5 -7.9 +6.9 +10.1 -4.4 +6.9 +10.4 -15.8 +13.5 -12.2 -0.9
Steps
(reduced) 		31
(6) 		49
(24) 		62
(12) 		72
(22) 		80
(5) 		87
(12) 		93
(18) 		98
(23) 		103
(3) 		107
(7) 		111
(11)
Approximation of harmonics in 25ed7/4 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +16.1 +4.0 +0.8 +5.4 +16.7 -4.8 +17.9 +6.6 -0.4 -3.4 -2.8 +1.0
Relative (%) +41.5 +10.4 +2.2 +13.9 +43.0 -12.3 +46.2 +17.0 -0.9 -8.8 -7.4 +2.5
Steps
(reduced) 		115
(15) 		118
(18) 		121
(21) 		124
(24) 		127
(2) 		129
(4) 		132
(7) 		134
(9) 		136
(11) 		138
(13) 		140
(15) 		142
(17)
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